The term "mass" really isn't being used for different things, but rather one must respect the particular physical framework in order to deduce what it means.
When we can neglect general relativity, then the energy of a system is defined as the conserved quantity associated with time-translation invariance. In flat space, translations in time are (almost) always symmetries of the field theories that describe elementary particles. If translations in space are also symmetries, then we can also define momentum. The mass of a system can then be defined as the value of the energy when the total momentum is zero. To be precise, this is the "inertial" mass of the system.
For an elementary particle, the mass can be identified as a parameter in the equation of motion for the corresponding quantum field. It is this parameter that is directly related to the Higgs mechanism in the Standard Model. For a composite system, the energy of the system involves not only the masses of the elementary constituents, but also the potential energy of the configuration. For example, the ground state of the hydrogen atom has a binding energy of -13.6 eV, so the mass of the hydrogen atom is
$$ m_H = m_p + m_e - 13.6~\mathrm{eV}/c^2,$$
i.e., a bit less than the sum of the masses of its parts.
Now, when we refer to gravitational mass, we're usually talking about the Newtownian limit of general relativity. In this case, the notion that the gravitational and inertial masses are identical is part of the collection of ideas known as the
equivalence principle. There is no proof of the equivalence principle in general, but the Standard Model of particle physics is consistent with it.
In general relativity proper, it is not just mass that acts as a source for gravity, but all forms of energy. In fact, the relevant object that appears in the Einstein field equations of GR is the stress-energy tensor. The stress-energy tensor is just a fancy way of expressing the conserved quantities associated with translations in time and space, which I said earlier were just the energy and and momentum. Note that having gravity couple to the energy is actually required from the observation that the mass of a bound state also involves a contribution from the potential energy of the system. It would not be consistent (in the Newtonian limit) for gravity to ignore this contribution.
Now, in a field theory describing elementary particles, like the Standard Model, or even some sort of string theory, every particle in the theory (not just the graviton) contributes to the stress tensor. So every particle, massive or massless, is a source for gravity and therefore spacetime curvature. It is therefore not true that only the graviton is responsible for geometry: any matter will curve the geometry around it.
The graviton is special, since it represents a localized excitation of the metric field. In fact, this manifests itself in the requirement that every other field must have a nonzero coupling to the graviton field. This is different from the other fundamental interactions, where an elementary particle must have a corresponding nonzero charge to participate in the interaction. For example, electrons and quarks are electrically charged, so they can participate in electromagnetic interactions involving photons. However, particles like neutrinos and the Higgs particle are electrically neutral, so they don't directly interact with photons.
Gravity and gravitons are different, since every particle has a nonzero gravitational charge corresponding to its energy. In a certain sense, this coupling to the graviton is one way to represent precisely how a given particle influences the geometry around it.
